Subgroups — #25104797

Order 11008.

738 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144940	C₂	1	1	yes	maximal, characteristic
2	#19143532	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143313	C₂	1	1	yes	maximal, characteristic
4	#19135572	C₂	1	1	yes	maximal, characteristic, complemented
5	#19135303	C₂	1	2	yes	maximal, complemented
6	#19135296	C₂	1	2	yes	maximal, complemented
7	#19134735	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134440	C₂	1	2	yes	maximal, complemented
9	#19134433	C₂	1	2	yes	maximal, complemented
10	C₄×#13063236	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1962	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063332	C₂²	1	2	yes	—
13	#14583456	C₂²	1	1	yes	characteristic
14	#14583267	C₄	1	2	yes	complemented
15	#14583263	C₄	1	2	yes	complemented
16	#14583258	C₄	1	4	yes	complemented
17	#14583028	—	2	4	no	—
18	#14582971	—	2	4	no	—
19	#14582927	C₂²	1	2	yes	—
20	#14582923	C₂²	1	1	yes	characteristic
21	#14582895	C₂²	1	2	yes	—
22	#14582895	C₂²	1	2	yes	—
23	#14582866	C₂²	1	2	yes	—
24	#14582857	C₂²	1	2	yes	—
25	C₂×#13063236	C₂²	1	1	yes	characteristic
26	C₂×#13063236	C₂²	1	1	yes	characteristic
27	C₂×#13063236	C₂²	1	1	yes	characteristic
28	C₂×#13063236	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
31	#14582663	C₂²	1	2	yes	—
32	#14582663	C₂²	1	2	yes	—
33	#14582659	C₂²	1	2	yes	—
34	#14582659	C₂²	1	2	yes	—
35	C₄₃×#532	C₂²	1	1	yes	characteristic
36	C₄₃×#423	C₂²	1	1	yes	characteristic
37	C₂×#104×C₄₃	C₂²	1	2	yes	—
38	C₂×#104×C₄₃	C₂²	1	2	yes	—
39	C₄³×C₄₃	C₂²	1	1	yes	characteristic
40	#13063332	—	2	4	no	—
41	#13063332	—	2	4	no	—
42	C₂×#10621877	—	2	2	no	—
43	C₂×#10621877	C₂³	1	2	yes	—
44	C₂²×C₄₃⋊C₈	—	2	2	no	—
45	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
49	#13063325	C₂×C₄	1	2	yes	—
50	#13063317	C₂×C₄	1	2	yes	—
51	#13063304	—	2	2	no	—
52	C₄×Dic₈₆	—	2	2	no	—
53	#13063284	C₂³	1	2	yes	—
54	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
55	Dic₄₃⋊C₈	—	2	4	no	—
56	#13063249	C₂×C₄	1	4	yes	—
57	#13063249	C₂×C₄	1	4	yes	—
58	#13063246	C₂×C₄	1	4	yes	—
59	#13063246	C₂×C₄	1	4	yes	—
60	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
61	#13063236	—	2	2	no	—
62	#13063236	—	2	2	no	—
63	#13063236	C₂×C₄	1	2	yes	—
64	#13063236	C₂×C₄	1	2	yes	—
65	#13063236	D₄	1	4	yes	—
66	#13063235	—	2	2	no	—
67	C₄×C₄₃⋊C₈	—	2	2	no	—
68	C₄×C₄₃⋊C₈	—	2	2	no	—
69	C₈×Dic₄₃	—	2	2	no	—
70	C₂×#41×C₄₃	C₂³	1	2	yes	—
71	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
73	#127×C₄₃	C₂×C₄	1	1	yes	characteristic
74	#124×C₄₃	C₂×C₄	1	2	yes	—
75	#121×C₄₃	—	2	2	no	—
76	C₄×D₄×C₄₃	—	2	2	no	—
77	C₂×C₄²×C₄₃	—	2	2	no	—
78	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₄₃	—	2	2	no	—
82	#104×C₄₃	—	2	4	no	—
83	#104×C₄₃	—	2	4	no	—
84	#41×C₄₃	—	2	2	no	—
85	#41×C₄₃	—	2	2	no	—
86	#41×C₄₃	—	2	2	no	—
87	#41×C₄₃	—	2	2	no	—
88	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₄₃	—	2	2	no	—
90	C₂×D₄×C₄₃	—	2	2	no	—
91	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₄₃	—	2	2	no	—
93	C₂²×C₄×C₄₃	—	2	1	no	—
94	C₂²×C₄×C₄₃	—	2	1	no	—
95	C₂²×C₄×C₄₃	#41	1	2	yes	—
96	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂×Dic₈₆	—	2	2	no	—
98	M₁₆×C₄₃	—	2	2	no	—
99	M₁₆×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₂²⋊C₄×C₄₃	—	2	4	no	—
103	C₄²×C₄₃	—	2	1	no	—
104	C₄²×C₄₃	—	2	1	no	—
105	C₄²×C₄₃	—	4	4	no	—
106	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	2	yes	—
109	C₄²×C₄₃	#41	1	2	yes	—
110	C₄²×C₄₃	#41	1	2	yes	—
111	C₂²⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
114	Dic₄₃⋊C₄	—	2	4	no	—
115	C₄×Dic₄₃	—	2	2	no	—
116	#10621877	—	2	2	no	—
117	#10621877	—	2	2	no	—
118	#10621877	—	4	4	no	—
119	C₂×C₄₃⋊C₈	—	2	1	no	—
120	C₂×C₄₃⋊C₈	—	2	1	no	—
121	C₂×C₄₃⋊C₈	—	2	1	no	—
122	C₂×C₄₃⋊C₈	—	2	1	no	—
123	C₂×C₄₃⋊C₈	—	2	1	no	—
124	C₂×C₄₃⋊C₈	—	2	1	no	—
125	C₂×C₄₃⋊C₈	—	2	2	no	—
126	C₂×C₄₃⋊C₈	—	4	4	no	—
127	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
129	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
130	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
131	C₂×C₄₃⋊C₈	#41	1	2	yes	—
132	C₂×C₄₃⋊C₈	#41	1	2	yes	—
133	C₂³×C₄₃	—	2	2	no	—
134	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
135	Q₈×C₄₃	—	2	2	no	—
136	Q₈×C₄₃	—	2	2	no	—
137	D₄×C₄₃	—	2	2	no	—
138	D₄×C₄₃	—	2	2	no	—
139	D₄×C₄₃	—	4	2	no	—
140	D₄×C₄₃	—	4	2	no	—
141	D₄×C₄₃	—	8	2	no	—
142	C₂×C₄×C₄₃	—	2	4	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
144	C₂×C₄×C₄₃	—	2	2	no	—
145	C₂×C₄×C₄₃	—	2	2	no	—
146	C₂×C₄×C₄₃	—	4	2	no	—
147	C₂×C₄×C₄₃	—	4	2	no	—
148	C₂×C₄×C₄₃	—	4	2	no	—
149	C₂×C₄×C₄₃	—	4	2	no	—
150	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₄₃	—	2	4	no	—
159	Dic₈₆	—	8	2	no	—
160	C₈×C₄₃	—	4	2	no	—
161	C₄₃⋊C₈	—	2	2	no	—
162	C₄₃⋊C₈	—	2	2	no	—
163	C₄₃⋊C₈	—	2	2	no	—
164	C₄₃⋊C₈	—	4	2	no	—
165	C₄₃⋊C₈	#124	1	4	yes	—
166	#21662	—	43	1	no	maximal, Sylow
167	C₂²×C₄₃	—	2	2	no	—
168	C₂²×C₄₃	—	4	4	no	—
169	C₂²×C₄₃	—	4	2	no	—
170	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₄₃	#529	1	1	yes	characteristic
172	C₂²×C₄₃	#529	1	1	yes	characteristic
173	C₄×C₄₃	—	2	4	no	—
174	C₄×C₄₃	—	2	2	no	—
175	C₄×C₄₃	—	2	2	no	—
176	C₄×C₄₃	—	4	2	no	—
177	C₄×C₄₃	—	4	2	no	—
178	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
179	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
180	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
181	C₄×C₄₃	#532	1	1	yes	characteristic
182	Dic₄₃	—	4	4	no	—
183	#3135	—	43	1	no	—
184	#2961	—	43	1	no	—
185	#2932	—	43	1	no	—
186	#1994	—	43	1	no	—
187	#1962	D₄₃	1	1	yes	characteristic
188	#1938	—	43	2	no	—
189	#1937	—	43	2	no	—
190	#1838	—	43	1	no	—
191	#1765	—	43	2	no	—
192	#1764	—	43	2	no	—
193	C₄×#97	—	43	1	no	—
194	C₂×C₄₃	—	2	1	no	—
195	C₂×C₄₃	—	2	1	no	—
196	C₂×C₄₃	—	8	2	no	—
197	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
198	C₂×C₄₃	#2956	1	2	yes	—
199	C₂×#131	—	43	2	no	—
200	#533	—	43	1	no	—
201	#532	C₂×D₄₃	1	1	yes	characteristic
202	#491	—	43	2	no	—
203	#490	—	43	2	no	—
204	#489	—	43	4	no	—
205	#446	—	86	4	no	—
206	#435	—	86	4	no	—
207	#424	—	43	2	no	—
208	#423	—	43	1	no	—
209	#423	C₂×D₄₃	1	1	yes	characteristic
210	C₂×#104	C₂×D₄₃	1	2	yes	—
211	C₂×#104	C₂×D₄₃	1	2	yes	—
212	#416	—	43	2	no	—
213	#416	—	43	2	no	—
214	#408	—	43	2	no	—
215	#405	—	43	2	no	—
216	C₂×#97	—	43	1	no	—
217	C₂×#97	—	43	1	no	—
218	C₂×#97	—	43	1	no	—
219	C₂×#97	—	43	1	no	—
220	C₂×C₄×C₈	—	43	1	no	—
221	C₂×C₄×C₈	—	43	1	no	—
222	C₄³	C₂×D₄₃	1	1	yes	characteristic
223	#341	—	43	2	no	—
224	#341	—	43	2	no	—
225	#339	—	43	2	no	—
226	#339	—	43	2	no	—
227	C₄₃	#21662	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	C₂²×D₄₃	1	2	yes	—
229	#131	—	86	4	no	—
230	#131	—	86	4	no	—
231	C₂×M₁₆	—	43	2	no	—
232	C₂×M₁₆	—	86	2	no	—
233	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
234	C₂²×C₈	—	43	1	no	—
235	C₂²×C₈	—	43	1	no	—
236	C₂²×C₈	—	43	1	no	—
237	C₂²×C₈	—	43	1	no	—
238	C₂²×C₈	—	86	2	no	—
239	C₄⋊Q₈	—	43	2	no	—
240	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
241	#127	C₄×D₄₃	1	1	yes	characteristic
242	#125	—	43	2	no	—
243	#124	C₄×D₄₃	1	2	yes	—
244	C₂²⋊Q₈	—	86	2	no	—
245	#121	—	2	2	no	—
246	C₄×Q₈	—	86	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	—	43	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	86	4	no	—
255	C₄⋊C₈	—	86	2	no	—
256	#104	—	2	4	no	—
257	#104	—	2	4	no	—
258	#103	—	43	4	no	—
259	#103	—	43	4	no	—
260	#102	—	43	4	no	—
261	#102	—	43	4	no	—
262	C₂²⋊C₈	—	86	4	no	—
263	#97	—	43	2	no	—
264	#97	—	43	2	no	—
265	#97	—	43	4	no	—
266	#97	—	86	2	no	—
267	#97	—	86	2	no	—
268	#97	—	86	2	no	—
269	C₄×C₈	—	86	2	no	—
270	C₄×C₈	—	86	2	no	—
271	C₄×C₈	—	86	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	C₂×Q₈	—	86	2	no	—
277	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	#10621900	1	2	yes	—
285	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	86	2	no	—
289	M₁₆	—	86	2	no	—
290	M₁₆	—	172	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	43	2	no	—
293	C₂×C₈	—	43	2	no	—
294	C₂×C₈	—	43	2	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	43	2	no	—
298	C₂×C₈	—	86	1	no	—
299	C₂×C₈	—	86	1	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	86	1	no	—
302	C₂×C₈	—	86	1	no	—
303	C₂×C₈	—	86	1	no	—
304	C₂×C₈	—	86	2	no	—
305	C₂×C₈	—	172	4	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	43	4	no	—
308	C₄⋊C₄	—	86	4	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	86	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	4	4	no	—
315	C₄²	—	86	2	no	—
316	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
318	C₄²	D₄×D₄₃	1	2	yes	—
319	C₄²	#10621900	1	2	yes	—
320	C₄²	#10621900	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	C₂×#10621900	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	344	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	86	4	no	—
340	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
341	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
345	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
347	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
348	C₈	—	4	2	no	—
349	C₈	—	43	4	no	—
350	C₈	—	86	2	no	—
351	C₈	—	86	2	no	—
352	C₈	—	86	2	no	—
353	C₈	—	172	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
358	C₂²	#14583430	1	1	yes	characteristic
359	C₂²	#14583430	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	172	4	no	—
366	C₄	C₄×#10621900	1	1	yes	characteristic
367	C₄	C₄×#10621900	1	1	yes	characteristic
368	C₄	#124×D₄₃	1	1	yes	characteristic
369	C₄	#14583447	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	#19137142	1	1	yes	characteristic
374	C₂	#19143501	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1962	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₄₃×#532	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1962	D₄₃
hypocenter	C₄₃	#21662
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #19143532	C₂	semidirect
4. #19135572	C₂	semidirect
5. #19135303	C₂	semidirect
6. #19135296	C₂	semidirect
7. #19134735	C₂	semidirect
8. #19134440	C₂	semidirect
9. #19134433	C₂	semidirect
10. C₄×#13063236	C₂	semidirect
14. #14583267	C₄	semidirect
15. #14583263	C₄	semidirect
16. #14583258	C₄	semidirect
227. C₄₃	#21662	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19144940	2
#19143532	2
#19143313	2
#19135572	2
#19135303	2
#19135296	2
#19134735	2
#19134440	2
#19134433	2
C₄×#13063236	2
C₄₃×#1962	2
#21662	43